What is the vertex form of  y = 3x^2 − 50x+300 ?

Apr 22, 2018

$y = 3 {\left(x - \frac{25}{3}\right)}^{2} + \frac{275}{3}$

Explanation:

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{y = a {\left(x - h\right)}^{2} + k} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$

$\text{obtain this form using "color(blue)"completing the square}$

• " the coefficient of the "x^2" term must be 1"

$\text{factor out 3}$

$\Rightarrow y = 3 \left({x}^{2} - \frac{50}{3} x + 100\right)$

• " add/subtract "(1/2"coefficient of the x-term")^2" to"
${x}^{2} - \frac{50}{3} x$

$y = 3 \left({x}^{2} + 2 \left(- \frac{25}{3}\right) x \textcolor{red}{+ \frac{625}{9}} \textcolor{red}{- \frac{625}{9}} + 100\right)$

$\textcolor{w h i t e}{y} = 3 {\left(x - \frac{25}{3}\right)}^{2} + 3 \left(- \frac{625}{9} + 100\right)$

$\textcolor{w h i t e}{y} = 3 {\left(x - \frac{25}{3}\right)}^{2} + \frac{275}{3} \leftarrow \textcolor{b l u e}{\text{in vertex form}}$

Apr 22, 2018

The vertex form of equation is $y = 3 {\left(x - \frac{25}{3}\right)}^{2} + \frac{1100}{12}$

Explanation:

$y = 3 {x}^{2} - 50 x + 300 \mathmr{and} y = 3 \left({x}^{2} - \frac{50}{3} x\right) + 300$ or

$y = 3 \left\{{x}^{2} - \frac{50}{3} x + {\left(\frac{50}{6}\right)}^{2}\right\} - \frac{2500}{12} + 300$ or

$y = 3 {\left(x - \frac{25}{3}\right)}^{2} + \frac{1100}{12}$ Comparing with vertex form of

equation y = a(x-h)^2+k ; (h,k) being vertex we find

here $h = \frac{25}{3} , k = \frac{1100}{12} \therefore$ Vertex is at $\left(8.33 , 91.67\right)$

The vertex form of equation is $y = 3 {\left(x - \frac{25}{3}\right)}^{2} + \frac{1100}{12}$

graph{3 x^2-50 x+300 [-320, 320, -160, 160]} [Ans]